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Article Dans Une Revue Graphs and Combinatorics Année : 2022

Block Elimination Distance

Résumé

We introduce the parameter of {\sl block elimination distance} as a measure of how close a graph is to some particular graph class. Formally, given a graph class ${\cal G}$, the class ${\cal B}({\cal G})$ contains all graphs whose blocks belong to ${\cal G}$ and the class ${\cal A}({\cal G})$ contains all graphs where the removal of a vertex creates a graph in ${\cal G}$. Given a hereditary graph class ${\cal G}$, we recursively define ${\cal G}^{(k)}$ so that ${\cal G}^{(0)}={\cal B}({\cal G})$ and, if $k\geq 1$, ${\cal G}^{(k)}={\cal B}({\cal A}({\cal G}^{(k-1)}))$. We show that, for every non-trivial hereditary class ${\cal G}$, the problem of deciding whether $G\in{\cal G}^{(k)}$ is {\sf NP}-complete. We focus on the case where ${\cal G}$ is minor-closed and we study the minor obstruction set of ${\cal G}^{(k)}$ i.e., the minor-minimal graphs not in ${\cal G}^{(k)}$. We prove that the size of the obstructions of ${\cal G}^{(k)}$ is upper bounded by some explicit function of $k$ and the maximum size of a minor obstruction of ${\cal G}$. This implies that the problem of deciding whether $G\in{\cal G}^{(k)}$ is {\sl constructively} fixed parameter tractable, when parameterized by $k$. Finally, we give two graph operations that generate members of ${\cal G}^{(k)}$ from members of ${\cal G}^{(k-1)}$ and we prove that this set of operations is complete for the class ${\cal O}$ of outerplanar graphs.
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Dates et versions

hal-03835914 , version 1 (17-10-2023)

Identifiants

Citer

Öznur Yaşar Diner, Archontia Giannopoulou, Giannos Stamoulis, Dimitrios M. Thilikos. Block Elimination Distance. Graphs and Combinatorics, 2022, 38 (5), pp.#133. ⟨10.1007/s00373-022-02513-y⟩. ⟨hal-03835914⟩
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